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Events for February – June 2016

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Feb. 02, 2016 1pm (MATH 220)	Grad Algebra/Logic Agnes Szendrei (CU Boulder) Algebras with few subpowers, Part 2 --- cancelled due to snow
Feb. 02, 2016 1pm (Math 350)	Functional Analysis Organizational Meeting (University of Colorado) X We will have an organizational meeting for this seminar on Tuesday, Feb. 2 at 1 p.m. in Math 350. All are welcome!! Organizational Meeting Sponsored by the Meyer Fund
Feb. 02, 2016 2pm (MATH 220)	Peter Mayr Finitely generated varieties 3 -- canceled due to snow!
Feb. 03, 2016 10am (MATH 350…	Lauren Grimley (Texas A&M) X Hochschild cohomology of an associative algebra has a graded ring structure and a compatible graded Lie algebra structure, making it a Gerstenhaber algebra. The structures on Hochschild cohomology reveal information about the algebra and its deformations. One class of algebras with interesting Hochschild cohomology is quantum complete intersections which are a non-commutative analog of truncated polynomial rings. The behavior of Hochschild cohomology of this class of algebras has been shown to vary greatly based on the choice of quantum coefficients, exhibiting behaviors unseen with commutative algebras. In this talk, we explore the behavior of Hochschild cohomology of group extensions of quantum complete intersections. Hochschild cohomology of group extensions of quantum complete intersections Sponsored by the Meyer Fund
Feb. 03, 2016 11am (MATH 350…	Timo de Wolff (Texas A&M) X Deciding nonnegativity of real polynomials is a key question in real algebraic geometry with crucial importance in polynomial optimization. Since this problem is NP-hard, one is interested in finding sufficient conditions (certificates) for nonnegativity, which are easier to check. Since the 19th century the standard certificates are sums of squares (SOS); see particularly Hilbert's 17th problem. In this talk we introduce polynomials supported on circuits. For this class, nonnegativity is characterized by an invariant, which can be derived immediately from the initial polynomial. In consequence, we obtain an entirely new class of nonnegativity certificates, which are independent of SOS certificates. Our certificates crucially extend geometric programming approaches for the computation of lower bounds in polynomial optimization. Particularly, for polynomials with simplex Newton polytope our approach is significantly faster and often yields better than bounds than semidefinite programming, which is the standard method for polynomial optimization. These results generalize earlier works by Fidalgo, Ghasemi, Kovacec, Marshall, and Reznick. The talk is based on joint work with Sadik Iliman. New Certificates for Nonnegativity via Circuit Polynomials and Geometric Programming Sponsored by the Meyer Fund
Feb. 09, 2016 1pm (MATH 220)	Grad Algebra/Logic Agnes Szendrei (CU Boulder) Algebras with few subpowers, Part 2
Feb. 09, 2016 1pm (Math 350)	Functional Analysis Konrad Aguilar (University of Denver) X AF algebras are inductive limits of finite-dimensional C*-algebras. We will present that certain classes of unital AF algebras can be described as metric limits with respect to the quantum propinquity metric. To achieve this, we provide a quasi-Leibniz quantum compact metric space structure for unital AF algebras with faithful tracial state. With this structure available, we show that certain classes of AF algebras form continuous images of maps from suitable metric spaces to the topology induced by the quantum propinquity metric. These classes include UHF algebras, Effros-Shen AF algebras, and the Cantor AF algebra. Joint work with Dr. Frederic Latremoliere. Quantum propinquity convergence of AF algebras Sponsored by the Meyer Fund
Feb. 09, 2016 2pm (MATH 220)	Peter Mayr (CU) Finitely generated varieties 3
Feb. 09, 2016 2pm (MATH 350)	Lie Theory Nat Thiem (CU) X Combinatorial representation theory seeks to associate the representations of groups and algebras to combinatorial gadgets. For many groups, we know how to find combinatorics in dimensions of irreducible modules, index sets, and branching rules. However, a seemingly far more difficult problem seeks to construct an explicit basis for the irreducible modules that have nice formulas for the action of the group. This talk gives an overview of approaches, some examples of success, and some long-standing mysteries. Combinatorial module constructions
Feb. 09, 2016 3pm (Math 350)	Markus Pflaum (CU Boulder) Introduction to Stratified Spaces
Feb. 09, 2016 4pm (Math 220)	Jonathan Lamar (CU Boulder) X I will define a polynomial representation of GL(n,C) and show that there is an equivalence of categories between polynomial representations of GL(n,C) and finite-dimensional representations of the Symmetric group on n letters. Polynomial Representations of GL(n,C)
Feb. 10, 2016 4pm (MATH 350)	Grad Student Seminar Noah Williams X Traditional classroom formats, dominated by lecture and at the whims of content-cramming tempos, often stifle what we love most about mathematics and the skills mathematicians value. On the other hand, Inquiry-Based Learning (IBL) approaches introduce mathematics to students as a living, breathing organism, where success is measured by struggle, creativity, and hard work. Teachers and students act as facilitators and apprentices rather than dispensers and consumers, and in-class activities are student-centered, involving group-work and presentations. In this interactive session, we'll discuss IBL, and I'll share my experience from the IBL workshop I attended last summer. Participants will leave with a basic understanding of IBL and some ideas for implementing IBL practices in their classrooms. How do IBLong? An introduction to teaching mathematics with IBL methods
Feb. 16, 2016 12:10pm (MATH …	Kempner Patricio Gallardo (University of Georgia) X We will describe new and old ways to understand the moduli space of points and lines in the plane. Depending on our choice of the allowed degenerations, we can obtain a moduli space that is arbitrarily singular or a configuration space that is smooth with well behaved boundary. For the most part, this talk will be expository, though if time allows we will describe a connection to my recent work with Noah Giansiracusa and Kenny Ascher. Point and line to plane
Feb. 16, 2016 1pm (MATH 220)	Grad Algebra/Logic Agnes Szendrei (CU Boulder) The ultimate generalization of Gaussian elimination to solve CSPs, Part 2
Feb. 16, 2016 2pm (MATH 350)	Lie Theory Shawn Burkett (CU) X Under certain conditions, the action of a finite group on a finite group induces an action of on the set of supercharacter theories of . In particular, this is possible when acts via automorphisms. The two trivial supercharacter theories must be fixed under these actions, but it is not so clear if any other fixed points exist. In this talk, such actions will be discussed, with particular focus on determining groups in which no nontrivial supercharacter theories of are fixed by and those in which all supercharacter theories are fixed by . Actions on the set of supercharacter theories of a finite group
Feb. 16, 2016 3pm (Math 350)	Algebraic Geometry Patricio Gallardo (UGA) X TBA TBA Sponsored by the Meyer Fund
Feb. 16, 2016 4pm (Math 220)	Megan Ly (CU Boulder) X I will introduce classical Schur-Weyl duality and discuss other situations where Schur-Weyl duality arises. Schur-Weyl duality
Feb. 17, 2016 4pm (MATH 350)	Grad Student Seminar Sebastian Bozlee X A Galois connection is a relationship between posets following the same pattern as that between fixed fields and Galois groups in Galois theory. This talk will be an introduction to Galois connections via examples you've probably seen before, but didn't realize were Galois connections. Missed Connections
Feb. 23, 2016 1pm (Math 350)	Functional Analysis Sooran Kang (University of Colorado, Boulder) X Quantum Heisenberg manifolds (QHMs) are noncommutative manifolds constructed by Marc Rieffel as a strict deformation quantization of Heisenberg manifolds (in the direction of the Poisson bracket). They are the fibers of a continuous field of noncommutative -algebras , where are parameters of the Poisson bracket, is a positive integer and is the deformation parameter. The QHM is in fact a generalized fixed point algebra of a certain crossed product -algebra by construction and it can be also realized as a crossed product by a Hilbert -bimodule, Cuntz-Pimsner algebra and twisted groupoid -algebra. The goal of this series of lectures is to introduce various types of -algebras and to discuss the structure of the QHM in some details. In the first lecture, we briefly discuss the history and the motivation of deformation quantization, and introduce the QHMs as a strict deformation quantization of Heisenberg manifolds. Quantum Heisenberg Manifolds
Feb. 23, 2016 1pm (MATH 220)	Grad Algebra/Logic Julie Linman (CU Boulder) Minimal functions on the random permutation, Part 1
Feb. 23, 2016 2pm (MATH 350)	Lie Theory Erica Shannon (CU) Some stuff about invariant forms
Feb. 23, 2016 3pm (Math 350)	Markus Pflaum (CU Boulder) Introduction to Stratified Spaces (Part II)
Feb. 23, 2016 4pm (Math 220)	Megan Ly (CU Boulder) X We will explore the rich combinatorial structure based on set partitions that arises from the supercharacter theory of unipotent upper triangular matrix groups. Supercharacters and Set Partition Combinatorics
Feb. 24, 2016 4pm (MATH 350)	Grad Student Seminar Robert Hines X Have any of your students approached you and said something to the effect of, "I saw this guy on youtube say 1+2+3+4+...=-1/12..."? Perhaps this talk will prepare you for the next time (no guarantees). The plan is to present one or two proofs of the functional equation relating \zeta(s) and \zeta(1-s). As a byproduct, we obtain special values of the zeta function, including \zeta(-1)=-1/12. We'll start with the definition of the zeta function (and a word or two about why it's interesting), the concept of analytic continuation, and (time permitting) we'll touch upon the gamma function, Poisson summation, theta functions, crazy contour integrals, Bernoulli numbers, and so on. Some knowledge of complex analysis is presupposed. The functional equation of the Riemann zeta function
Feb. 25, 2016 12:10pm (MATH …	Kempner Antoine Julien (Norwegian University of Science and Technology) X In this talk, my goal is to give an introduction to some of the mathematics behind quasicrystals. Quasicrystals were discovered in 1982, when Dan Schechtmann observed a material which produced a diffraction pattern made of sharp peaks, but with a 10-fold rotational symmetry. This indicated that the material was highly ordered, but the atoms were nevertheless arranged in a non-periodic way. These quasicrystals can be modeled by certain aperiodic tilings, amongst which the famous Penrose tiling. What makes aperiodic tilings so interesting--besides their aesthetic appeal--is that they can be studied using tools from many areas of mathematics: combinatorics, topology, dynamics, operator algebras... While the study of tilings borrows from various areas of mathematics, it doesn't go just one way: tiling techniques were used by Giordano, Matui, Putnam and Skau to prove a purely dynamical statement: any Z^d free minimal action on a Cantor set is orbit equivalent to an action of Z. Quasiperiodic tilings and orbit equivalence of dynamical systems
Mar. 01, 2016 11am (MATH 350)	Number Theory Anna Haensch (Duquesne University) X There is a well known correspondence between lattices and codes via the classical “construction A.” With this, the weight enumerator for codes corresponds to the theta series for lattices, where one counts the number of codewords by composition, and the other counts the number of vectors in a lattice of a certain length. In this talk, we will explore how some of the attendant machinery of theta series are born out in this correspondence. In particular, we will consider the Kneser-Hecke operator, a code theoretic analogue of the classical Hecke operator. Kneser-Hecke operators for codes over finite chain rings
Mar. 01, 2016 1pm (MATH 220)	Grad Algebra/Logic Julie Linman (CU Boulder) Minimal functions on the random permutation, Part 2
Mar. 01, 2016 2pm (MATH 350)	Lie Theory Jay Taylor (University of Padova) X The classical groups , , , have long been of interest in group theory; often because of their close relationship to finite simple groups. These groups are all examples of finite reductive groups. Such a group is obtained as the fixed point subgroup of a connected reductive algebraic group under a Frobenius endomorphism . Assuming the characteristic of the field defining is not too small (greater than 5 suffices) then Kawanaka has defined a family of complex representations of the finite group called Generalised Gelfand—Graev Representations (GGGRs). Using these representations Kawanaka conjectured that one could associate to every irreducible character of a well-defined -stable unipotent conjugacy class of which he called the wave front set. Under certain assumptions on the underlying characteristic Lusztig proved Kawanaka’s conjecture, asserting the existence of the wave front set. In this talk we will give an introduction to these ideas an present our result that Kawanaka’s conjecture holds whenever the GGGRs are defined. Generalised Gelfand—Graev Representations and Wave Front Sets Sponsored by the Meyer Fund
Mar. 01, 2016 3pm (** Math 2…	Jordan Watts (CU Boulder) X A famous question of Dusa McDuff, often referred to as the "McDuff Conjecture", is whether there exists a non-Hamiltonian symplectic circle action with isolated fixed points on a compact symplectic manifold. Susan Tolman recently answered this question in the affirmative, constructing a -dimensional such space with exactly fixed points. A crucial ingredient to this construction involves Hamiltonian circle actions on complex manifolds and orbifolds in which the interaction between the complex structure and the symplectic form is fairly weak. Specifically, versions of Sjamaar's holomorphic slice theorem, the birational equivalence theorem of Guillemin and Sternberg, as well as reduction, cutting, and blow-up (all of which work in the Kaehler world) are required in this weaker setting. All of these theorems and constructions are extended to this weaker setting in joint work by Tolman and myself. In particular, by weak, we mean that if is the vector field induced by the circle action, the symplectic form, and the complex structure, then on the complement of the fixed point set. This condition is sufficient for all of the theorems and constructions above except for the blow-up (which also requires tameness at the point to be blown-up). In this talk, I will focus on the holomorphic slice theorem about a fixed point, reduction, and (time-permitting) the birational equivalence theorem proven in the joint paper. Tame Circle Actions
Mar. 08, 2016 11am (MATH 350)	Number Theory Joseph H. Silverman (Brown University) X A divisibilty sequence is a sequence of integers W_1,W_2,... with the property that if m divides n, then W_m divides W_n. Classical divisibility sequences include a^n-b^n, Fibonacci and Lucas sequences, and elliptic divisibility sequences. These tend to grow rapidly, and their behavior modulo p is quite interesting. After surveying these sequences, I will move on to higher dimensional analogues such as gcd(a^n-1,b^n-1), which tend to grow slowly, and faster growing Divisor Divisibility Sequences that Katherine Stange, Patrick Ingram, and I have recently been studying. Despite their elementary-seeming nature, divisibility sequences arise in many areas of mathematics due to their intrinsic definition in terms of arithmetic intersections on arithmetic groups. Along the way I will describe many open questions whose easy statements belie their depth. Divisibility Sequences
Mar. 08, 2016 12:10pm (MATH …	Kempner Paul Horn (University of Denver) X Graphs are a mathematical abstraction of networks, which on their most basic level capture objects and their relationships. Although graphs are discrete objects, there turns out to be many striking analogies between them and Riemannian manifolds. Most famous of these are the relationships between the spectra of the Laplace operator on graphs and the spectra of the Laplace-Beltrami operator on a manifold. Recent work, however, has attempted to bring other ideas from the world of geometry into the graph setting – notably by trying to introduce notions of curvature on graphs. In this talk, I will describe some of my work in this area and how it reflects a rich interplay between the local structure of a graph, the behavior of random walks on graphs, and more global geometric properties of a graph. The Geometry of Graphs
Mar. 08, 2016 1pm (MATH 220)	Grad Algebra/Logic Andrew Maurer (University of Georgia) X Lie superalgebras can be defined abstractly as an associative ZZ/(2)-graded algebra with the usual commutator operation. It turns out the bracket structure respects the grading. When studying the representation theory of Lie superalgebras, we may use the tools provided by support varieties. This involves taking Spec of the cohomology ring modulo its nilradical. As in the theory of Lie algebras, our study begins with complex Lie superalgebras. Quickly, we encounter a problem in that the cohomology ring is finite dimensional as a complex vector space, and thus support varieties are points. This may be remedied by using Hochschild's relative cohomology, which have much more interesting, and much more elusive, support varieties. In this talk I will speak about some results concerning so-called classical Lie superalgebras, with particular emphasis on cohomology relative to its even Lie subsuperalgebra. (Relative) Cohomology and Support Varieties and Lie Superalgebras
Mar. 08, 2016 2pm (MATH 350)	Lie Theory Marcelo Aguiar (Cornell) X Consider an arrangement of linear hyperplanes integral with respect to a given lattice. The lattice gives rise to a torus and the arrangement to a subdivision of the torus. We are interested in the combinatorics of this subdivision. We will describe questions and results for particular lattices associated to root systems and arrangements associated to graphs. This is joint work in progress with Swee Hong Chan. Integral hyperplane arrangements Sponsored by the Meyer Fund
Mar. 08, 2016 4pm (Math 220)	Shawn Burkett (CU Boulder) X A central idea in group theory is to study a group by studying homomorphic images into families of groups in which a lot of information is already known. For example, given a group action, one gets a homomorphism of the acting group into the symmetric group on the elements of the set being acted on. The permutation representation so acquired may contain significant information about the acting group. Extending this idea, if a group G acts on a k-vector space V, one gets a homomorphism of G into GL(V), called a linear representation. That is, each element of G acts as an invertible linear transformation on V. Linear transformations have many invariants, one being the trace. If we compose the linear representation with the trace map, one obtains a map from G to k, called a k-character. In this talk, we will discuss the basics of linear representation theory and character theory in great detail, providing all necessary information to introduce supercharacter theories of finite groups. Character Theory of Finite Groups
Mar. 09, 2016 3pm (Math 350)	João Nuno Mestre (Utrecht University) X We explain how an extension of Haefliger's approach to transverse measures for foliations allows us to define and study measures and geometric measures (densities) on differentiable stacks. The abstract theory works for any differentiable stack, but it becomes very concrete for proper stacks - for example, when computing the volume associated with a density, we recover the explicit formulas that are taken as definition by Weinstein. This talk is based on joint work with Marius Crainic. Measures and densities on differentiable stacks Sponsored by the Meyer Fund
Mar. 09, 2016 4:15pm (MATH 3…	Grad Student Seminar Natalie Coston X The fifteen puzzle consists of a 4x4 puzzle board grid on which there are 15 tiles and one blank. The goal of the puzzle it to slide the tiles around until the numbers appearing on the board are ordered 1 to 15. If we started with an ordered board and were mixing the tiles back up to give to a friend, as curious mathematicians we might wonder "How do I know if my board is really mixed?" or, "How long should I mix these tiles?" In this talk, which is based off the paper "The Mixing Time of the Fifteen Puzzle" by Morris and Raymer, I will introduce Markov Chains and demonstrate how they can be used to bound the time it takes to mix the 15 puzzle. The Math Behind the Mixing of the 15 Puzzle
Mar. 10, 2016 3pm (Weber 201…	Algebraic Geometry Joe Silverman (Brown) X TBA TBA
Mar. 11, 2016 4pm (MATH 350)	Grad Student Seminar Sarah Salmon X This atypical time for Slow Pitch is part of Welcome Weekend! Come learn some neat stuff and meet some prospective students. We will explore some connections between simple graphs and polynomial rings. Generally speaking, finding monomial prime decompositions of ideals or minimal generating sets of ideals in polynomial rings can be rather unpleasant. If you start with a simple graph, we can construct some interesting ideals using vertices (called the cover ideal) or edges (called the edge ideal). Using properties of the graph, finding minimal generating sets of cover ideals or minimal monomial prime decompositions of the edge ideal is not so bad! The short version: We're going to build some ideals, find some nice generating sets, and draw some pictures (as always)! Simple Graphs and Monomial Ideals: How I Spent Summer 2014
Mar. 15, 2016 1pm (Math 350)	Functional Analysis Sooran Kang (University of Colorado, Boulder) X Abstract: As mentioned in the first lecture, the quantum Heisenberg manifold can be realized as a crossed product by a Hilbert -bimodule which is the special case of Cuntz-Pimsner algebras. In this lecture (and the next lecture), we introduce Cuntz-Pimsnef algebras and crossed products by Hilbert -modules, and discuss how we realize the quantum Heisenberg manifold as those algebras using the classical Picard groups. Cuntz-Pimsner algebras and crossed products by -bimodules. Sponsored by the Meyer Fund
Mar. 15, 2016 1pm (MATH 220)	Grad Algebra/Logic Dmitriy Zhuk (Moscow State Unuversity) X The standard way to parameterize interesting subclasses of the constraint satisfaction problem (CSP) is via finite relational structures. The main problem is to classify those subclasses that are tractable (solvable in polynomial time) and those that are NP-complete. It was conjectured that if a core of a relational structure has a weak near-unanimity polymorphism then the corresponding constraint satisfaction problem is tractable, otherwise it is NP-complete. This conjecture was proved by Andrei Bulatov for 3-element domain, by Petar Markovic for 4-element domain. We present the proof of this conjecture for 5-element domain. Precisely, we prove the remaining part of this conjecture showing that a CSP instance can be solved in polynomial time if the relational structure has a weak near-unanimity polymorphism. The proof of CSP Dichotomy Conjecture for 5-element domain Sponsored by the Meyer Fund
Mar. 15, 2016 2pm (MATH 220)	Dmitriy Zhuk (Moscow State University) X The main problem in clone theory is to get a description of all clones. In 1941 Emil Post obtained an amazing description of all clones on 2 elements, but in 1959 it was proved that there exists a continuum of clones on k elements if k > 2. The first part of my talk is devoted to the efforts to describe uncountable lattices of clones, in particular, I show how uncountable lattice of all clones of self-dual operations on 3 elements can be described. However, obtaining such description for other maximal clones seems to be too cumbersome and complicated. This motivated me to focus on the following questions: how to check properties of a clone and how to find finite sublattices. Despite the fact that clone theory has been studied for many years, we still don't know how to check simple properties of clones. For example we don't know how to check whether a clone defined by a relation is finitely generated, or how to decide how many clones contain a given operation. In the second part of the talk I try to collect such open problems and present some partial results. Open problems in clone theory Sponsored by the Meyer Fund
Mar. 15, 2016 3pm (Math 350)	Christopher Seaton (Rhodes College) X Let be a compact Lie group and a unitary representation of . Treating as a real symplectic manifold and choosing the homogeneous quadratic moment map, we can form the symplectic quotient, the quotient where is the zero fiber of the moment map. The space has a variety of structures, including that of a symplectic stratified space and a semialgabraic set, and its algebra of smooth functions has a graded subalgebra of regular functions defined as a quotient of the real polynomial invariants of the -action on . In this talk, we will discuss the use of the polynomial invariants of the -representation to construct symplectomorphisms and distinguish between symplectic quotients. In particular, we will study the problem of which symplectic quotients are symplectomorphic or diffeomorphic to symplectic orbifolds. We demonstrate that in some cases, has many of the nice properties of a ring of orbifold invariants and describe explicit computations of invariants of . Along the way, we will see some recent results in computational invariant theory that are relevant in other contexts. (Joint work with Joshua Cape, Carla Farsi, Hans-Christian Herbig, Daniel Herden, Ethan Lawler, and Gerald Schwarz) Constructing symplectomorphisms between singular linear symplectic quotients Sponsored by the Meyer Fund
Mar. 15, 2016 4pm (Math 220)	Shawn Burkett (CU Boulder) X Supercharacter theory was originally developed to describe the representation theory of algebra groups, a family of groups in which unimodular groups belong. Essentially, supercharacter theories are certain types of central subalgebras which approximate the usual character theory of a finite group. In this talk, we will introduce supercharacter theories from multiple persepectives and discuss a few ways to use the structure of a given finite group to construct nontrivial supercharacter theories. Time permitting, I will discuss some recent work of myself and Jonathan Lamar (CU Boulder) on induced actions on the set of supercharacter theories of a finite group. Random talk on a supercharacter theory
Mar. 17, 2016 12:10pm (MATH …	Kempner Guoliang Yu (Texas A&M) X I will explain how certain geometric properties of groups can be used to study rigidity and non-rigidity of manifolds. In particular, I will discuss how a certain secondary invariants of elliptic operators can be used to estimate the degree of non-rigidity of manifolds. Parts of the work are joint with Erik Guentner, Romain Tessera, Shmuel Weinberger, and Zhizhang Xie. This talk will be accessible to graduate students. Geometry of groups and rigidity of manifolds
Mar. 17, 2016 3pm (MATH 350)	Probability Asad Lodhia (MIT) X We provide an overview of results on the Central Limit Theorem for Linear Statistics of Eigenvalues. We will discuss how the central limit theorem depends on the scale at which the eigenvalues are analyzed and mention various results. We discuss how universality of the behavior of the Law of Large Numbers (e.g. local semicircle law) relates to universality of the Central Limit Theorem for these statistics. CLTs for Random Matrix Eigenvalues
Mar. 28, 2016 4pm (Math 350)	Functional Analysis Daniel Goncalves (Universidade Federal de Santa Catarina) X In this talk we show how to obtain representations of graph algebras from branching systems and show how these representations connect to the Perron-Frobenius operator from Ergodic Theory. We also describe a sufficient condition that guarantees faithfulness of our representations and show how to use these representations to give a new proof to the converse of the Cuntz-Krieger Uniqueness Theorem for graph algebras (including Leavitt path algebras). Representations of graph algebras via branching systems Sponsored by the Meyer Fund
Mar. 29, 2016 11am (MATH 350)	Number Theory Pedro Berrizbeitia (Ulam Visitor from Universidad Simon Bolivar) X Let an odd prime and a non quadratic residue mod . Let be the ring of integers of . If is coprime to , then has order in the multiplicative group , hence must be congruent mod to (why?). Using properties of the classical biquadratic residue symbol including the Biquadratic Reciprocity Law we present two different methods to compute the sign, which leads to what we call the "Norm Relation Theorem". As an application we prove the following rather curious property of Mersenne primes that we believe had not been noticed: Let be prime. Let be prime (i.e. a Mersenne prime). It is known that has integer solutions . Moreover, is unique with . We prove that !! Examples: ; and ; and The Biquadratic Symbol in . A new property of the Mersenne Primes
Mar. 29, 2016 1:10pm (MATH 3…	Kempner Daniel Goncalves (Universidade Federal de Santa Caterina) X Symbolic dynamics is a fundamental area of dynamical systems, and is typically concerned with a finite alphabet , spaces of infinite sequences or , and subspaces of or which are closed under the shift map (shift spaces). Over the years many approaches have been proposed to generalize shift spaces to the infinite alphabet case, one of the hurdles to overcome is the fact that the countable product of an infinite discrete space is not locally compact. In this talk I will describe a recent approach to infinite shift spaces introduced by Ott, Tomforde and Willis. I intend to show that, while many of the finite alphabet symbolic dynamics results still hold for these infinite alphabet shifts (as for example the connection of conjugacy of edge shifts of graphs with isomorphism of the associated C*-algebras), some surprising results appear (for example there are (M+1)-step shifts not conjugate to an M-step shift). One sided shift spaces over infinite alphabets
Mar. 29, 2016 4pm (Math 220)	Sarah Salmon (CU Boulder) TBD
Mar. 30, 2016 4pm (Math 220)	Functional Analysis Piotr Hajac (IMPAN) X Representing Z/NZ as roots of unity, we restrict a natural U(1)-action on the Heegaard quantum 3-sphere to Z/NZ, and call the quotient spaces Heegaard quantum lens spaces. Then we use this representation of Z/NZ to construct an associated complex line bundle. The main result is the stable non-triviality of these line bundles over any of the quantum lens spaces we consider. We use the pullback structure of the C*-algebra of the lens space to compute its K-theory via the Mayer-Vietoris six-term exact sequence. Then we combine an explicit form of the odd-to-even connecting homomorphism (Milnor idempotent) with Chern-Galois theory (strong connections) to prove the stable non-triviality of the bundles. (Joint work with Adam Rennie and Bartosz Zielinski.) THE K-THEORY OF HEEGAARD QUANTUM LENS SPACES Sponsored by the Meyer Fund
CANCELED Mar. 31, 2016	Probability Hakima Bessaih (University of Wyoming at Laramie) X We study Brinkman's equations with microscale properties that ar ehighly heterogeneous in space and time. The time variations are controlled by a stochastic particle dynamics described by an SDE. Our main results include the derivation of macroscale equations and showing that the macroscale equations are deterministic. We use the asymptotic properties of the SDE and the periodicity of the Brinkman's coefficient in the space variable to prove the convergence result. The SDE has a unique invariant measure that is ergodic and strongly mixing. The macro scale equations are derived through an averaging principle of the slow motion (fluid velocity) with respect to the fast motion (particle dynamics) and also by averaging the Brinkman's coefficient with respect to the space variable. Our results can be extended to more general nonlinear diffusion equations with heterogeneous coefficients. This is a joint work with Yalchin Efendiev and Florian Maris. HOMOGENIZATION OF BRINKMAN FLOWS IN HETEROGENEOUS DYNAMIC MEDIA
Apr. 01, 2016 1:10pm (ECCR 1…	Kempner Giulia Sacca (Stony Brook) X There are very few known examples of compact Kahler manifolds that admit a holomorphic symplectic form. Two series of examples were discovered by Beauville in the 80's and then two sporadic (or exceptional) examples of dimension 6 and 10 were found by O'Grady. In this talk I will review the construction of these examples and describe some recent work describing the geometry of the two exceptional examples. One way to achieve this is to use Lagrangian fibrations. I will review what a Lagrangian fibration is as well as its basic properties, and then show how this structure allows one to study or construct examples of holomorphic symplectic manifolds. Holomorphic symplectic manifolds and Lagrangian fibrations Sponsored by the Meyer Fund
Apr. 01, 2016 3pm (ECCR 265)	Kempner John Hunter (University of California Davis) X Surface plasmons (SPs) are electromagnetic surface waves that propagate on the interface between an insulator and a conductor, such as air and gold. At short wavelengths, the frequency of SPs approaches a nonzero constant independent of their wavenumber, and we consider the effect of nonlinearity on SPs in this nondispersive limit. We show that the amplitude of weakly nonlinear SPs satisfies a cubically nonlinear, nonlocal asymptotic equation. We discuss the existence of smooth and weak solutions of this equation, and the effects of nonlinearity on the focusing and possible formation of singularities in SPs. This is joint work with Ryan Halabi. Nonlinear surface plasmons
Apr. 05, 2016 1pm (MATH 220)	Grad Algebra/Logic Agnes Szendrei (CU Boulder) The ultimate generalization of Gaussian elimination to solve CSPs, Part 3
Apr. 05, 2016 1pm (Math 350)	Functional Analysis Sooran Kang (University of Colorado, Boulder) X We first discuss the original construction of the QHMs by Rieffel as generalized fixed point algebras of certain crossed product -algebras in some detail. Then we describe the QHM as a crossed product by a Hilbert -bimodule, , where is the fixed point algebra of the QHM and is the first spectral subspace of the QHM. Quantum Heisenberg manifolds as crossed products by Hilbert -bimodules. Sponsored by the Meyer Fund
Apr. 05, 2016 4pm (Math220)	Katharine Adamyk (CU Boulder) X In part II of this talk, we will examine several other definitions for Catalan numbers, along with the various explicit formulas for Cn that these definitions lead to. In addition, we will discuss some applications, generalizations, and interesting properties of Catalan numbers. Even More Things to Count with Catalan Numbers
Apr. 08, 2016 4pm (MATH 100)	Kempner R. Tyrell Rockafellar (University of Washington) X Uncertainty over the future, even if well supported by statistical information, causes a fundamental difficulty in optimization. Decisions can't pin future "costs" to specific values and are instead limited only to shaping their probability distributions as random variables. Hard upper bounds on "costs" may then be impossible or too expensive to enforce. On the other hand, traditional approaches in such a contest in terms of safety margins built around standard deviation, or probability of failure based on quantiles, can have serious mathematical shortcomings. It is essential to build optimization models in which cost variables are kept "adequately" below some threshold. Adequacy can be expressed by a choice of how risk is quantified, but that choice should fit with axioms of "coherency". Questions arise then on the statistical side as well. In estimating the consequences of decisions, through regression or some other means, the handling of databases may need to interact with the particular approach taken to risk. Risk, Optimization and Statistics
Apr. 12, 2016 11am (MATH 350)	Number Theory James Ricci (Daemen College) X One of the many interesting algebraic objects associated to a given elliptic curve defined over the rational numbers, E / Q, is its full-torsion representation ρ_E:Gal( Q /Q) → GL₂(Ẑ). Generalizing this idea, one can create another full-torsion Galois representation, ρ_(E₁,E₂): Gal( Q /Q)→(GL₂(Ẑ))² associated to a pair (E₁,E₂) of elliptic curves defined over Q. The goal of this talk is to provide an infinite number of concrete examples of pairs of elliptic curves whose associated full-torsion Galois representation ρ_(E₁,E₂) has maximal image. Elliptic Curves with Maximally Disjoint Division Fields
Apr. 12, 2016 12:10pm (MATH …	Kempner Sandra Laursen (Ethnography & Evaluation Research (E&ER) at CU Boulder) X Mathematics serves as students’ gateway to majors and careers in STEM fields (science, technology, engineering, mathematics), but lack of success in their initial college math courses deters many STEM-intending students from progressing in their desired pathways. While studies across the STEM disciplines support the use of student-centered teaching and learning to increase student learning and persistence, there has been less previous research on these approaches in college mathematics. Our group’s study of inquiry-based learning (IBL) as implemented in over 100 course sections at four universities shows that student-centered approaches can succeed when implemented on a departmental scale (http://www.colorado.edu/eer/research/steminquiry.html ). Despite variation in how IBL was implemented, student outcomes are improved in IBL courses relative to traditionally taught courses, as assessed by general measures that apply across course types. Particularly striking, the use of IBL eliminates a sizable gender gap that disfavors women students in lecture-based courses. Other evidence suggests particular benefits to initially lower-achieving students, and to pre-service K-12 teachers. The study supports IBL teaching as a means to improve learning outcomes and, potentially, student retention in college mathematics. In this session, I will highlight key results from our work that mathematics instructors may find useful to consider and to share with students and colleagues. Student outcomes of inquiry-based learning (IBL) in undergraduate mathematics: Key findings from education research
Apr. 12, 2016 1pm (Math 350)	Functional Analysis Sooran Kang (University of Colorado, Boulder) X As promised in the first lecture, we conclude this series of Ulam lectures by describing the quantum Heisenberg manifolds as twisted groupoid -algebras. First we discuss the definition of a twist groupoid over a groupoid associated to a local homeomorphism and a locally compact Hausdorff space, and discuss the result by Deaconu-Kumjian-Muhly, which says that the twisted groupoid -algebra, , is isomorphic to a Cuntz-Pimsner algebra , where is a Hilbert -correspondence. Combining this result with the result of the isomorphism classes of the QHMs, we show that the QHM can be realized as a twisted groupoid -algebra. Quantum Heisenberg manifolds as twisted groupoid -algebras Sponsored by the Meyer Fund
Apr. 12, 2016 1pm (MATH 220)	Grad Algebra/Logic Nik Ruskuc (University of St Andrews) X Let V be a class of algebras (usually a variety), and let P be an algebraic property. Is it true that the direct product AxB of two algebras from V satisfies the property P if and only if A and B satisfy P? If not, what are the conditions on A and B that make AxB satisfy P? For instance, in the group case the first question has a positive answer for most standard properties P (such as, say, finite generation, finite presentability, residual finiteness, etc.) Peter Mayr and I have been pursuing a project of investigating these questions within the context of Universal Algebra. It turns out that group theoretic results often generalise to much broader classes of algebra, e.g. Malcev varieties and such like. However, it would appear that semigroups by and large tend to remain outside the scope of our general theorems. My aim for this talk is to present it as a kind of 'companion' to our work, in which I will describe the curious landscape for semigroups, and present some remaining open problems. Finiteness properties for direct products of semigroups Sponsored by the Meyer Fund
Apr. 12, 2016 2pm (MATH 220)	Nik Ruskuc (University of St Andrews) X Magnus' solution for the word problem (1932) for one-relator groups is one of the first results in the then-emerging field of combinatorial group theory, and the corresponding problem for one-relation semigroups is perhaps the most easily stated and longest-standing open problem in combinatorial algebra. This talk will be mostly historical in nature, with the aim of outlining the relationship between the two problems, and the present state of knowledge about one-relation semigroups. I will also indicate some possible ways forward, which I am currently pursuing in a project with Robert Gray. One relation groups and semigroups Sponsored by the Meyer Fund
Apr. 12, 2016 4pm (Math220)	Megan Ly (CU Boulder) X Elementary particles such as electrons have an intrinsic angular momentum called "spin". In order to account for spin, we must use the double cover SU(2) of the three dimensional rotation group SO(3) to describe the rotations of fundamental quantum systems. In this talk we construct the irreducible representations of using first and second order differential operators to classify different spin states. Spin and irreducible representations of
Apr. 13, 2016 4pm (MATH 350)	Grad Student Seminar Leo Herr X What is a union? Marital, setwise, or otherwise, they're all types of colimits. Enter the hermeneutic circle hand-in-hand with Yoneda, learn a family recipe for triangles, and discover crossword puzzles are colimits, too! We'll be extremely concrete in discussions of unions, coequalizers, colimits, the Yoneda Lemma, Simplicial objects, sheaves, and ultimately topoi. Colimits Aren't Stupid -- A Geometric Realization
Apr. 14, 2016 12:10pm (MATH …	Kempner Paul Baum (Penn State University) X Topics in the talk are : #1. The basic definition of K-theory #2. A brief history of K-theory #3. Algebraic versus topological K-theory #4. The unity of K-theory This is an expository talk intended for a general mathematical audience. What is K-theory and what is it good for?
Apr. 14, 2016 1pm (Math 350)	Noncomm Geometry Paul Baum (Penn State University) X Topics in the lecture are : #1. Two low-dimensional examples #2. Dirac Operator #3. Spin-c manifolds #4. Proof of the Dirac case #5. The Dirac case implies the general case The aim of this expository talk is to give a clear and simple proof of Atiyah-Singer as a corollary of Bott periodicity. It is my intention that this proof should be understandable to graduate students. ATIYAH-SINGER REVISITED Sponsored by the Meyer Fund
Apr. 19, 2016 1pm (MATH 220)	Grad Algebra/Logic Jakub Bulin (CU Boulder) Finitely related algebras, Part 1
Apr. 19, 2016 3pm (Math 350)	Laura Scull (Fort Lewis College) X Effective orbifolds were originally defined via charts and atlases, analogous to manifolds. In the current literature, they are often defined instead via certain topological groupoids, which gives a more convenient category for studying them. These two approaches are known to be equivalent and produce the same objects. More recently, applications arising in geometry and physics have led to an interest in more general ineffective orbifolds. However, the current generalizations of the atlas definition do not correspond to the objects created by ineffective topological groupoids. I will discuss a project, joint with D. Pronk and M. Tommasini, where we develop an alternate atlas definition for ineffective orbifolds. Our atlases generalize the existing effective atlas definition, and our definition leads to the same objects as those defined by ineffective topological groupoids. Atlases for Ineffective Orbifolds Sponsored by the Meyer Fund
Apr. 20, 2016 4pm (MATH 350)	Grad Student Seminar Hanson Smith X Quadratic reciprocity was first proved by Gauss, and in Disquisitions Arithmeticae he calls it the "Fundamental Theorem." In this talk, we will motivate quadratic reciprocity with some of the ideas and conjectures of Fermat, Euler, and Legendre. I will present a proof that is relatively short and requires only a familiarity with groups, rings, and fields to understand. The goal of this talk is to give an appreciation for a useful and historically significant theorem, so I aim for it to be very accessible. A Proof of Quadratic Reciprocity
Apr. 21, 2016 3pm (MATH 350)	Probability Diane Holcomb (University of Arizona) X The Gaussian Unitary/Orthogonal Ensembles (GUE/GOE) are some of the most studied Hermitian random matrix models. They may be generalized to a one parameter family of point processes indexed by beta called the beta-Hermite ensembles. When appropriately rescaled, the eigenvalues in the interior of the spectrum converge to a translation invariant limiting point process called the Sine_beta process. One expects the Sine_beta process to have a number of points that is roughly the length of the interval times a fixed constant (the density of the process). We find the asymptotic probability of two rare events. The first is a large deviation problem for the density of points in a large interval. The second is the asymptotic probability of overcrowding in a fixed interval. Rare events for the Sine_beta process
Apr. 26, 2016 12:10pm (MATH …	Kempner David Aristoff (Colorado State University) X We discuss Markov chain sampling problems arising in computational chemistry. In particular, we show how to sample functions of paths, especially hitting times, in the presence of practical difficulties like metastability and slow convergence. We show how a mathematical framework for these techniques allows for quantification of convergence, opening the way for improvement in efficiency and minimization of error. Sampling distributions of Markov chains
Apr. 26, 2016 1pm (MATH 220)	Grad Algebra/Logic Jakub Bulin (CU Boulder) Finitely related algebras, Part 2
Apr. 26, 2016 1pm (Math 350)	Functional Analysis Nadia Larsen (University of Oslo) X In recent work with Laca, Neshveyev, Sims and Webster we investigate the factor types of the extremal KMS states for the preferred dynamics on the -algebra of a strongly connected finite k-graph. At inverse temperature 1, the prevalent outcome is a type III factor. We compute its Connes invariant in terms of the spectral radii of the coordinate matrices and the degree of cycles in the graph. Our analysis has two main steps: one is to analyze the von Neumann algebra of the equivalence relation of the -graph groupoid, the other is to formulate and prove a Frobenius-type result for the family of matrices associated to the -graph. Von Neumann algebras of strongly connected higher-rank graphs Sponsored by the Meyer Fund
Apr. 26, 2016 3pm (Math 350)	Dorette Pronk (Dalhousie University) X For this talk we will consider orbifolds as Lie groupoids with certain properties with generalized maps between them. The generalized maps are obtained by taking the bicategory of fractions of the 2-category of orbifold groupoids with groupoid homomorphisms. The fractions are needed because we want to consider Morita equivalences as invertible We will first consider what generalized maps between orbifolds are and how we can think of them in terms of a generalization of an atlas refinement together with a map on the refinement. We will then discuss the structure of the 2-cells between them (the arrows of the mapping groupoid) and find that this is simpler than what one would expect for an arbitrary bicategory of fractions, and we will give a nice geometric interpretation. We will end by calculating various examples, and in particular showing that the two ineffective groupoids that Laura Scull presented last week are indeed not Morita equivalent. Mapping Spaces for Orbifolds Sponsored by the Meyer Fund
Apr. 27, 2016 4pm (MATH 350)	Grad Student Seminar Elizabeth Parsons X Digital signal processing (DSP) is the numerical manipulation of continuous time, continuous amplitude sinusoidal functions, usually with the intention of discretizing analog signals. In this talk, we will discuss the importance of the discrete Fourier transform, short time Fourier transform and wavelet transform in DSP. Our focus will be on the time-frequency representations obtained from the short time Fourier transform and wavelet transform. We will find hidden images within audio files which can only be obtained using time-frequency representations of signals. Digital Signal Processing: Using Math to Turn Music into Pictures
May. 17, 2016 11am (MATH 350)	Number Theory Amanda Tucker (University of Rochester) X What's the difference between 3, 1/3, the sum from n=1 to infinity of 1/n^3, and pi? We will start with a discussion of what it means for a number to be integral, rational, algebraic, or transcendental. We will talk about the history of the multiple zeta values (certain real numbers) and some open problems about them. You will leave this talk knowing more about the real numbers than you thought was possible, but also with more questions about the real numbers than you thought was possible! Multiple zeta values
May. 17, 2016 1pm (MATH 350)	Algebraic Geometry Amanda Tucker (University of Rochester) X We will introduce the area of arithmetic statistics and the Cohen- Lenstra heuristics. We will give some background about the work of Bhargava- Shankar-Tsimerman and Taniguchi-Thorne on the Davenport-Heilbronn theorems. We will then explain our recent proof that 96.2% of totally real cubic fields have genus number one and, if time permits, talk about some applications. This represents joint work with Kevin McGown. The arithmetic statistics of number fields
Jun. 01, 2016 2pm (Math 350)	Functional Analysis Sarah Reznikoff (Kansas State University) X In 1948 Naimark showed that the C-algebra of compact operators, K(H), has a unique irreducible representation and asked whether the converse is true; that is, must any C-algebra with this property be isomorphic to K(H)? We will out line the proof of Akemann and Weaver (2003) that the existence of a counterexample to Naimark's Problem is consistent with the axioms of set theory ZFC. Notes on the relative consistency of Naimark's Problem Sponsored by the Meyer Fund

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